The completely antisymmetric solution ${\ensuremath{\Psi}}^{A}$ to the problem of the scattering of a fermion by a finite system of identical fermions is studied by means of the expansion ${\ensuremath{\Psi}}^{A}=\ensuremath{\Sigma}{\ensuremath{\phi}}_{\ensuremath{\alpha}}{\ensuremath{\psi}}_{\ensuremath{\alpha}}$, where ${{\ensuremath{\phi}}_{\ensuremath{\alpha}}}$ is a complete set of antisymmetric states for the target and ${\ensuremath{\psi}}_{\ensuremath{\alpha}}$ are one-particle functions. Coupled equations for the ${\ensuremath{\psi}}_{\ensuremath{\alpha}}$ are found that obey the proper boundary conditions. This is done by means of the integral equation for ${\ensuremath{\Psi}}^{A}$ and the use of projection operators. The elastic-scattering (optical-model) wave function ${\ensuremath{\psi}}_{0}$ is shown to obey an inhomogeneous differential equation, rather than a Schr\"odinger equation. The homogeneous solution is identical to the elastic wave function obtained when the projectile is distinguishable, while the inhomogeneous solution is due entirely to exchange effects. The function ${\ensuremath{\psi}}_{0}$ is identical to the optical-model wave function found by Bell and Squires, who showed that ${\ensuremath{\psi}}_{0}$ obeys a Schr\"odinger equation with an optical potential containing direct and exchange contributions. It is shown that ${\ensuremath{\psi}}_{0}$ yields the exact elastic amplitude including direct and exchange contributions, and a phase-shift analysis of the exchange term is given. The more standard form of solution ${\ensuremath{\Psi}}^{A}=\ensuremath{\Sigma}\mathcal{a}{{\ensuremath{\phi}}_{\ensuremath{\alpha}}{f}_{\ensuremath{\alpha}}}$, where $\mathcal{a}$ is an antisymmetrizer, is briefly discussed. The extension to the cases of inelastic scattering and deuteron elastic scattering is made.The completely antisymmetric solution ${\ensuremath{\Psi}}^{A}$ to the problem of the scattering of a fermion by a finite system of identical fermions is studied by means of the expansion ${\ensuremath{\Psi}}^{A}=\ensuremath{\Sigma}{\ensuremath{\phi}}_{\ensuremath{\alpha}}{\ensuremath{\psi}}_{\ensuremath{\alpha}}$, where ${{\ensuremath{\phi}}_{\ensuremath{\alpha}}}$ is a complete set of antisymmetric states for the target and ${\ensuremath{\psi}}_{\ensuremath{\alpha}}$ are one-particle functions. Coupled equations for the ${\ensuremath{\psi}}_{\ensuremath{\alpha}}$ are found that obey the proper boundary conditions. This is done by means of the integral equation for ${\ensuremath{\Psi}}^{A}$ and the use of projection operators. The elastic-scattering (optical-model) wave function ${\ensuremath{\psi}}_{0}$ is shown to obey an inhomogeneous differential equation, rather than a Schr\"odinger equation. The homogeneous solution is identical to the elastic wave function obtained when the projectile is distinguishable, while the inhomogeneous solution is due entirely to exchange effects. The function ${\ensuremath{\psi}}_{0}$ is identical to the optical-model wave function found by Bell and Squires, who showed that ${\ensuremath{\psi}}_{0}$ obeys a Schr\"odinger equation with an optical potential containing direct and exchange contributions. It is shown that ${\ensuremath{\psi}}_{0}$ yields the exact elastic amplitude including direct and exchange contributions, and a phase-shift analysis of the exchange term is given. The more standard form of solution ${\ensuremath{\Psi}}^{A}=\ensuremath{\Sigma}\mathcal{a}{{\ensuremath{\phi}}_{\ensuremath{\alpha}}{f}_{\ensuremath{\alpha}}}$, where $\mathcal{a}$ is an antisymmetrizer, is briefly discussed. The extension to the cases of inelastic scattering and deuteron elastic scattering is made.

Physical ReviewNov 22, 196521 citations

The Schrödinger Eigenvalue Problem for N-Particle Systems

W. Hunziker

Springer eBooksJan 1, 19778 citations

N Completeness, N Representability, and Geminal Expansions

Mary Beth Ruskai

The complete solution is presented for the $N$ -completeness problem, i.e., the problem of determining when a set of $p$ -particle functions generates symmetric or antisymmetric functions without applying a symmetrizing or antisymmetrizing operator. Then it is shown that a symmetric or antisymmetric function has finite 2 rank if, and only if, it has finite 1 rank. Finally, the relation between $N$ representability of $N$ completeness is discussed.The complete solution is presented for the $N$-completeness problem, i.e., the problem of determining when a set of $p$-particle functions generates symmetric or antisymmetric functions without applying a symmetrizing or antisymmetrizing operator. Then it is shown that a symmetric or antisymmetric function has finite 2 rank if, and only if, it has finite 1 rank. Finally, the relation between $N$ representability of $N$ completeness is discussed.

Physical Review AMar 1, 197214 citations

Radiation conditions and scattering theory for N-particle Schrödinger operators

Dimitri R. Yafaev

Springer eBooksApr 7, 20080 citations

The Quantum Distribution Functions

Michael J.R. Hoch

Consider a pair of particles that may be either two fermions or two bosons. e Schrödinger equation for the system is written in terms of the Hamiltonian operator for the system as H( , ) ( , ) ( , ),1 2 1 2 1 2c c= E (12.1) where 1 represents the coordinates of particle 1 and 2 the coordinates of particle 2. e Hamiltonian (1, 2) must be symmetric under particle interchange because of the indistinguishability of the particles, giving H H( , ) ( , ).1 2 2 1= (12.2) e pair wave function χ(1, 2) may be either symmetric or antisymmetric. is can be seen by introducing the permutation operator P12, which permutes the coordinates of the particles.Consider a pair of particles that may be either two fermions or two bosons. e Schrödinger equation for the system is written in terms of the Hamiltonian operator for the system as H( , ) ( , ) ( , ),1 2 1 2 1 2c c= E (12.1) where 1 represents the coordinates of particle 1 and 2 the coordinates of particle 2. e Hamiltonian (1, 2) must be symmetric under particle interchange because of the indistinguishability of the particles, giving H H( , ) ( , ).1 2 2 1= (12.2) e pair wave function χ(1, 2) may be either symmetric or antisymmetric. is can be seen by introducing the permutation operator P12, which permutes the coordinates of the particles.

CRC Press eBooksApr 19, 20160 citations

On the discrete spectrum of the Hamiltonian of an n-particle quantum system

M. A. Antonets, G. M. Zhislin, Ilya A. Shereshevskii

Theoretical and Mathematical PhysicsAug 1, 197323 citations

\mathcal{a}

is an antisymmetrizer, is briefly discussed. The extension to the cases of inelastic scattering and deuteron elastic scattering is made.

The complete solution is presented for the

N

-completeness problem, i.e., the problem of determining when a set of

p

-particle functions generates symmetric or antisymmetric functions without applying a symmetrizing or antisymmetrizing operator. Then it is shown that a symmetric or antisymmetric function has finite 2 rank if, and only if, it has finite 1 rank. Finally, the relation between

N

representability of

N

completeness is discussed.